Intereting Posts

To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I – A)$?
Forcing names, parameters in definitions, and the Iterative Conception of Set
An Example of a Nested Decreasing Sequence of Bounded Closed Sets with Empty Intersection
Polynomial equations in $n$ and $\phi(n)$
Equivalence of $a \rightarrow b$ and $\lnot a \vee b$
Quotient of the ring of integers of a number field by a prime ideal
Number of triangles inside given n-gon?
Notation: is it correct to state $3a=a3$?
Cardinality of a vector space versus the cardinality of its basis
A puzzle about integrability
Mental Math Techniques
Question related to pseudoprimes and Carmichael numbers
Seeking elegant proof why 0 divided by 0 does not equal 1
How to integrate $\int \frac{e^x dx}{1\,+\,e^{2x}}$
If $a$ is a real number then is $\sqrt{a^2}$ equal to $a$ or plus minus $a$?

The recursion formula

$$\sum_{k=0}^{n-1}{n\choose k}B_k=0$$

which is equation (34) from the MathWorld page is the most basic bernoulli number recursion.

- Power Series With Bernoulli Numbers
- Continued fraction expansion related to exponential generating function
- Bernoulli Number analog using Cosine
- Prove the von Staudt-Clausen congruence of the Bernoulli numbers
- Ways to prove Eulers formula for $\zeta(2n)$
- Very accurate approximations for $\sum\limits_{n=0}^\infty \frac{n}{a^n-1}$ and $\sum\limits_{n=0}^\infty \frac{n^{2m+1}}{e^n-1}$

Another recursion is given by the following:

Consider

$$\mathbf{A} = \begin{bmatrix}

-\frac{1}{2} & -\frac{1}{6} & -\frac{1}{12} & -\frac{1}{20} & -\frac{1}{30} & -\frac{1}{42}\\

\frac{1}{1} & 0 & 0 & 0 & 0 & 0\\

0 & \frac{1}{2} & 0 & 0 & 0 & 0\\

0 & 0 & \frac{1}{3} & 0 & 0 & 0\\

0 & 0 & 0 & \frac{1}{4} & 0 & 0\\

0 & 0 & 0 & 0 & \frac{1}{5} & 0 \end{bmatrix}$$

This matrix was constructed by putting $-1/(1\cdot2), -1/(2\cdot3), -1/(3\cdot4), …$ on the first row and $1/1, 1/2, 1/3, …$ on the subdiagonal.

Now with $k!A^ke$ with $e:=[1, 0, 0, 0, 0, 0]^T$ and $k < 6$ we get the coefficients of the kth Bernoulli polynomial. With the constant term in the ‘first’ (leftmost) position. For example

$$3!A^3e = [0, 1/2, -3/2, 1, 0, 0]^T$$

and

$$4!A^4e = [-1/30, 0, 1, -2, 1, 0]^T$$

Together with this equation from wikipedia

$$B_n(x) = \sum_{k=0}^n{n\choose k}B_kx^{n-k}$$

we are given another bernoulli number recursion.

If these two recursions are related – in what way exactly?

- Sum with binomial coefficients: $\sum_{k=1}^m \frac{1}{k}{m \choose k} $
- closed form expression for the sum of the first s items of alternating binomial coefficients
- Inductive Proof for Vandermonde's Identity?
- Show that $r_k^n/n \le \binom{kn}{n} < r_k^n$ where $r_k = \dfrac{k^k}{(k-1)^{k-1}}$
- Asymptotics of binomial coefficients and the entropy function
- Binomial coefficient problem
- Generating function for binomial coefficients $\binom{2n+k}{n}$ with fixed $k$
- Exercise from Comtet's Advanced Combinatorics: prove $27\sum_{n=1}^{\infty }1/\binom{2n}{n}=9+2\pi \sqrt{3}$
- Eigenvalues of a matrix with binomial entries
- Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$

Perhaps the thoughts which I followed recently give a satisfactory answer. I’ve looked at the “ZETA”-matrix and fiddled out the connection to the integral-representation in the Euler-MacLaurin-Formula. The article is not yet ready, needs some brushing and completing references and such. But as it might be helpful here: here is the link the integral in Euler-MacLaurin in connection with the Pascal-matrix

[update]: added the re-translation into bernoulli-numbers in the final formulae in the text

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